The depth, on the other hand, is not so simple to calculate. However, it can also be deciphered from the associated word in the following manner. As explained earlier, a branch within the \((D+1)\) -ary tree corresponds to a nested sequence of elementary \(D\) -balls within the melonic \(D\) -ball.
As a result of this nesting, the path of minimal graph distance leading from this internal vertex to the initial internal vertex lies within this sequence of elementary \(D\) -balls. However, the connectivity of the graph causes its value to deviate form that of the tree depth; the insertion of a \(D\) -ball does not mean that the new vertex is necessarily further away from the root vertex. In fact, consider a nested sequence of elementary \(D\) -balls and their associated internal vertices. Pick out the last elementary \(D\) -ball inserted. Say it has color \(i\)  and assume that its associated internal vertex is the first internal vertex in the sequence that is at depth \(r\) . All other internal vertices in that sequence have depth less than \(l\) .Internal vertices inserted earlier certainly do not have depth greater than \(r\) . Now, within this latest \(D\) -ball, insert an arbitrary nested sequence containing at least one elementary \(D\) -ball of each color except some color \(j \ne i\) . It emerges that the internal vertices associated to this newly inserted sequence all have depth \(r\) . Now insert a elementary \(D\) -ball of color \(j\) . Its associated internal vertex has depth \(r+1\) . We do not elaborate further on the reasoning here. However, the interested reader can convince oneself quite readily by drawing out the example given here or by looking in [1]}.
